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Session S30 - Mathematical Methods in Quantum Mechanics

Friday, July 16, 19:30 ~ 19:55 UTC-3

Semiclassical resolvent bound for long range Lipschitz potentials

Jacob Shapiro

University of Dayton, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We present an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2\Delta + V(x) -E$ in dimension $n\neq 2$, where $h, E >0$. The potential is real-valued, $V$ and $\partial_r V$ exhibit long range decay at infinity, and may grow like a sufficiently small negative power of $r$ as $r\to\infty$. The resolvent norm grows exponentially in $h^{ −1}$, but near infinity it grows linearly. When $V$ is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{−1/2}$ for some $C > 0$. This $E$-dependence is sharp and answers a question of Datchev and Jin.

Joint work with Jeffrey Galkowski (University College London).

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